Question

3. Let x be a rational number and y be an irrational number. Is x+y
necessarily an irrational number? Give an example in support of
answer.
​

Answer 1

Answer:

The sum of a rational and irrational number is always irrational.

Step-by-step explanation:

From Toppr:

Assume that a is rational, b is irrational, and a+b is rational. Since a and a+b are rational, we can write them as fractions.

Let a=dc​ and a+b=nm​

Substituting a=dc​ in a+b=nm​ gives the following:

dc​+b=nm​

Now, let's subtract dc​ from both sides of the equation.

b=nm​−dc​, or

b=nm​+(−dc​)

Since the rational numbers are closed under addition, b=nm​+(d−c​) is a rational number. However, the assumptions said that b is irrational, and b cannot be both rational and irrational. This is our contradiction, so it must be the case that the sum of a rational and an irrational number is always irrational.